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(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a82 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X20,15,21,16 X6,18,7,17 X14,19,15,20 X8,21,9,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code 4 10 12 16 2 22 18 20 6 14 8
A Braid Representative
A Morse Link Presentation K11a82 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a82/ThurstonBennequinNumber
Hyperbolic Volume 13.8883
A-Polynomial See Data:K11a82/A-polynomial

[edit Notes for K11a82's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 2

[edit Notes for K11a82's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-12 t^2+19 t-21+19 t^{-1} -12 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, -2 }
Jones polynomial -q^4+3 q^3-6 q^2+10 q-12+15 q^{-1} -15 q^{-2} +13 q^{-3} -10 q^{-4} +6 q^{-5} -3 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) -a^2 z^8+a^4 z^6-6 a^2 z^6+2 z^6+4 a^4 z^4-14 a^2 z^4-z^4 a^{-2} +9 z^4+5 a^4 z^2-15 a^2 z^2-3 z^2 a^{-2} +13 z^2+2 a^4-6 a^2-2 a^{-2} +7
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +7 a^4 z^8+11 a^2 z^8+3 z^8 a^{-2} +7 z^8+7 a^5 z^7-15 a z^7-7 z^7 a^{-1} +z^7 a^{-3} +5 a^6 z^6-14 a^4 z^6-43 a^2 z^6-12 z^6 a^{-2} -36 z^6+3 a^7 z^5-12 a^5 z^5-21 a^3 z^5-6 a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-5 a^6 z^4+14 a^4 z^4+51 a^2 z^4+15 z^4 a^{-2} +46 z^4-3 a^7 z^3+11 a^5 z^3+28 a^3 z^3+22 a z^3+13 z^3 a^{-1} +5 z^3 a^{-3} -a^8 z^2+a^6 z^2-7 a^4 z^2-27 a^2 z^2-8 z^2 a^{-2} -26 z^2-4 a^5 z-10 a^3 z-10 a z-6 z a^{-1} -2 z a^{-3} +2 a^4+6 a^2+2 a^{-2} +7
The A2 invariant q^{20}-q^{18}+2 q^{16}-q^{14}-q^{12}+q^{10}-4 q^8+2 q^6-2 q^4+2 q^2+3+3 q^{-4} - q^{-6} - q^{-12}
The G2 invariant q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+5 q^{106}-3 q^{104}-2 q^{102}+10 q^{100}-17 q^{98}+24 q^{96}-27 q^{94}+19 q^{92}-7 q^{90}-13 q^{88}+37 q^{86}-55 q^{84}+66 q^{82}-62 q^{80}+39 q^{78}-q^{76}-44 q^{74}+91 q^{72}-119 q^{70}+122 q^{68}-90 q^{66}+26 q^{64}+56 q^{62}-124 q^{60}+166 q^{58}-152 q^{56}+86 q^{54}+9 q^{52}-103 q^{50}+148 q^{48}-128 q^{46}+50 q^{44}+54 q^{42}-134 q^{40}+144 q^{38}-84 q^{36}-38 q^{34}+161 q^{32}-239 q^{30}+217 q^{28}-113 q^{26}-48 q^{24}+202 q^{22}-291 q^{20}+282 q^{18}-182 q^{16}+24 q^{14}+130 q^{12}-230 q^{10}+246 q^8-165 q^6+40 q^4+89 q^2-159+157 q^{-2} -70 q^{-4} -45 q^{-6} +148 q^{-8} -183 q^{-10} +139 q^{-12} -24 q^{-14} -108 q^{-16} +209 q^{-18} -228 q^{-20} +170 q^{-22} -58 q^{-24} -69 q^{-26} +156 q^{-28} -184 q^{-30} +152 q^{-32} -79 q^{-34} - q^{-36} +56 q^{-38} -81 q^{-40} +72 q^{-42} -47 q^{-44} +19 q^{-46} +3 q^{-48} -15 q^{-50} +14 q^{-52} -11 q^{-54} +6 q^{-56} -2 q^{-58} + q^{-60}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_116, K11a7, K11a33,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a33,}

Vassiliev invariants

V2 and V3: (0, 2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
0 16 0 0 16 0 -\frac{224}{3} -\frac{224}{3} -16 0 128 0 0 352 \frac{784}{3} -\frac{208}{3} \frac{128}{3} -64

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-2 is the signature of K11a82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           1-1
7          2 2
5         41 -3
3        62  4
1       64   -2
-1      96    3
-3     77     0
-5    68      -2
-7   47       3
-9  26        -4
-11 14         3
-13 2          -2
-151           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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